arX

iv:1

208.

5967

v1 [

hep-

ph]

29

Aug

201

2

ANL-HEP-PR-12-68

Combining QCD and electroweak corrections to dilepton

production in FEWZ

Ye Li1, 2, ∗ and Frank Petriello1, 2, †

1Department of Physics & Astronomy,

Northwestern University, Evanston, IL 60208, USA2High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA

Abstract

We combine the next-to-next-to-leading order (NNLO) QCD corrections to lepton-pair produc-

tion through the Drell-Yan mechanism with the next-to-leading order (NLO) electroweak correc-

tions within the framework of the FEWZ simulation code. Control over both sources of higher-order

contributions is necessary for measurements where percent-level theoretical predictions are crucial,

and in phase-space regions where the NLO electroweak corrections grow large. The inclusion of

both corrections in a single simulation code eliminates the need to separately incorporate such

effects as final-state radiation and electroweak Sudakov logarithms when comparing many exper-

imental results to theory. We recalculate the NLO electroweak corrections in the complex-mass

scheme for both massless and massive final-state leptons, and modify the QCD corrections in the

original FEWZ code to maintain consistency with the complex-mass scheme to the lowest order.

We present phenomenological results for LHC studies that include both NNLO QCD and NLO

electroweak corrections. In addition, we study several interesting kinematics features induced by

experimental cuts in the distribution of photon radiation at the LHC.

∗yeli2012@u.northwestern.edu†f-petriello@northwestern.edu

1

I. INTRODUCTION

The Drell-Yan (DY) production [1] of lepton pairs through the exchange of a Z-boson or

virtual photon plays a critical role at the LHC. The relative ease of identifying two leptons

leads to a very clean channel for detector calibration and performance studies [2]. The DY

process can serve as a luminosity monitor [3] because of its relatively large production rate,

and can be used to perform precision electroweak (EW) measurements [4] and constrain par-

ton distribution functions (PDFs) [5]. Moreover, the study of hadronic dilepton production

in the high invariant-mass region could reveal signs of new physics.

All of these uses require theoretical predictions at the percent level of precision. The

largest correction to the leading-order predictions comes from quantum chromodynamics

(QCD). QCD corrections to DY production up to the next-to-next-to-leading order (NNLO)

in the strong coupling constant have been previously calculated, both for the inclusive cross

section [6] and for differential quantities [7–12]. We have previously implemented the fully

differential corrections to the DY process in the form of the flexible parton-level simulation

code FEWZ (Fully Exclusive W and Z Production) [10, 13, 14]. Using FEWZ, predictions

for arbitrary kinematic distributions can be obtained, and for most such observables the

estimated theoretical uncertainty from higher-order QCD effects is a few percent. However,

at this level of precision the full EW correction at next-to-leading (NLO) cannot be neglected.

The NLO EW effects are known [15–20], and have been implemented in several public codes

such as ZGRAD2 [18, 19] and HORACE [15]. One major effect of the EW correction comes

from photon radiation from the final-state leptons, which can lead to large logarithmic

corrections sensitive to the lepton mass or calorimeter setting. The weak correction in the

high-energy Sudakov regime [21–24] can also cause a significant deviation from the leading

order contribution.

In this manuscript, we combine the NNLO QCD and NLO EW corrections to DY pro-

duction of lepton pairs in a new version of the FEWZ code. We rederive the EW one-loop

correction using the complex mass scheme (CMS). The higher-order contributions from both

the QCD and the EW theory are simply summed together to achieve O(α2s) +O(αEW ) ac-

curacy. In addition to providing the needed theoretical control over multiple sources of

higher-order corrections, the inclusion of EW effects in FEWZ also eliminates the need to

unfold photonic radiative effects before comparing data to theory, as is currently done in

LHC studies. For completeness, we also include the photon-induced dilepton production

channel at the lowest order in perturbation theory if the selected PDF set comes with a

photon distribution.

The features of the upgraded FEWZ are summarized below.

• The user can either choose from two hard-coded schemes for the input parameters,

the α(MZ) or Gµ scheme, or specify each coupling manually as in the original FEWZ.

However, if the user decides to manually input the coupling parameters, only the QED

2

corrections will be included in order to protect gauge invariance.

• Two different modes corresponding to either a zero or a non-zero lepton mass can be

chosen. Massless leptons lead to collinear divergences, and therefore photon-lepton

recombination must be applied when their separation drops below a certain detector

resolution limit. If no recombination procedure is performed and massive leptons are

selected, the predictions become sensitive to logarithms of the lepton mass.

• Histograms of photonics variables, such as the photon pT and photon-lepton separa-

tion, have been added. They can be specified in the histogram configuration file.

In order to demonstrate the features of the updated FEWZ, we present numerous phe-

nomenological results that can be compared to LHC data. We first check the results of our

calculation of the NLO EW corrections against previous results in the literature [20], and

find excellent agreement across a broad variety of observables. We then proceed to present

results for DY production at the LHC that demonstrate the interplay between QCD and

EW effects. We also study the effects of photon radiation in the DY process at the LHC,

and point out several interesting kinematic features that occur as a result of the imposed

experimental cuts. Our combination of fixed-order QCD at NNLO with the NLO EW cor-

rections is complementary to other efforts which combine NLO QCD plus parton-shower

effects with the EW corrections to W -boson production [25, 26].

This manuscript is organized as follows. In Section II, we present our conventions, discuss

different input parameter schemes, and introduce the complex-mass scheme for unstable

particles. In Section III, we discuss in detail our implementation of the complex mass

scheme and adapt the original QCD calculation to the complex-mass scheme. Numerical

results are presented in Section IV and compared to the previous literature. Histograms

of phenomenologically interesting observables at the LHC and the effect of different lepton

identification procedures are studied in detail. We conclude in Section V.

II. SETUP AND CONVENTIONS

Only three parameters are needed as basic inputs in the electroweak theory in addition

to the fermion masses. We study here two different input schemes, both of which use the

W and Z masses as the first two input parameters. The last input parameter is given

by either the Fermi constant in the Gµ scheme, or the effective electromagnetic coupling

at the Z mass in the α(MZ) scheme. At tree level, the effective electromagnetic coupling

can be derived from the Fermi constant as αGµ =√2GµM

2W (1 − M2

W/M2Z). We do not

consider the fine structure constant as an input-parameter option because it receives large

logarithmic corrections of the form α lnm2f , induced by light fermion masses in gauge boson

self-energy insertions. The effective coupling α(MZ) resums the above logarithms by running

the effective electromagnetic coupling from the scale Q = 0 to MZ . The Fermi constant Gµ

3

is derived from the effective theory describing the weak force in low energy processes. It

is most precisely measured in muon decay and receives a radiative correction denoted as

∆r, which contains ∆α(MZ). ∆r additionally contains ∆ρ, which accounts for the running

of the weak mixing angle and receives isospin-violating corrections induced by the heavy

top-quark mass. The Gµ input parameter scheme has been shown to be the choice most

stable against higher order EW corrections [20] and therefore is our default option here. We

list below the Standard Model parameters used in the updated FEWZ code:

Gµ = 1.16637× 10−5 GeV−2 ,

α(0) = 1/137.035999911 , α(MZ) = 1/128.91 ,

MZ,OS = 91.1876 GeV , ΓZ,OS = 2.4952 GeV ,

MW,OS = 80.403 GeV , ΓW,OS = 2.141 GeV ,

me = 5.1099891× 10−4 GeV , mµ = 0.105658369 GeV ,

mt = 172.9 GeV , mH = 125 GeV . (1)

The subscript OS denotes the on-shell values of the masses and widths. The fine structure

constant α(0) is only used for the photon-induced process. We note that the corrections are

insensitive to the choice of Higgs boson mass mH .

We begin our discussion of the complex-mass by rewriting the on-shell expressions for

the W and Z propagators in terms of the real and imaginary parts of their complex-plane

poles:1

s−M2V,OS + isΓV,OS/MV,OSθ(s)

=1

s−M2V + iΓVMV

(

1 +O(ΓV

MV

)

)

, (2)

where V stands for either the W -boson or Z-boson. We have assumed massless decay

products in writing the left-hand side of this equation. In the right-hand side we have

identified

MW/Z =MW/Z,OS

√

1 + Γ2W/Z,OS/M

2W/Z,OS

,ΓW/Z =ΓW/Z,OS

√

1 + Γ2W/Z,OS/M

2W/Z,OS

. (3)

The correct description of unstable particles produced on resonance is usually accomplished

through Dyson resummation of self-energy insertions. It unavoidably introduces a mixing of

perturbative orders and ruins gauge invariance if done incorrectly. It is especially tricky for

unstable particles running in loops, since the resonant term cannot be simply factored out

before the loop integration is performed. The complex mass scheme is a prescription that

consistently uses complex masses everywhere for unstable particles. The complex masses of

the W and Z gauge bosons are defined by

µ2W = M2

W − iMWΓW =M2

W,OS − iMW,OSΓW,OS

1 + Γ2W,OS/M

2W,OS

µ2Z = M2

Z − iMZΓZ =M2

Z,OS − iMZ,OSΓZ,OS

1 + Γ2Z,OS/M

2Z,OS

, (4)

4

where MW/Z,OS and ΓW/Z,OS are the mass and width of each boson respectively; their numer-

ical values are given in Eq. (1). The weak mixing angle is given by s2W = 1−c2W = 1−µ2W/µ2

Z ,

resulting in complex couplings of the fermions to the Z and W bosons. All relations follow-

ing from gauge invariance are preserved because the masses are only modified by analytic

continuation. Although the proof of unitarity order-by-order using the Cutkosky cutting

rule no longer holds, the spurious terms are of higher order in the perturbation series as

long as the width is a higher order object with respect to the mass. It should be noted that

unstable particles should not be used as external particles in the computation of S-matrix

elements [27].

There are other schemes which do not rely on complex parameters to cope with unstable

particles in the propagator, such as the on-shell (OS) scheme and the pole scheme (PS).

The OS scheme requires that a running width which vanishes at zero-momentum is kept

after Dyson resummation. Otherwise an artificial infrared divergence would appear from

the contribution of the would-be Goldstone boson in the case with finite fermion mass [28].

The OS scheme also requires the running width to be kept in the propagator during the loop

integration in order to describe the resonance correctly without spoiling gauge invariance.

The PS takes advantage of the fact that the residue on the resonance peak is gauge invariant.

The cross section in the pole scheme for the Z boson can be schematically written as:

dσ =Aresonant(s = M2

Z)

s−M2Z + iMZΓZ

+Aresonant(s)−Aresonant(s = M2

Z)

s−M2Z

+ Anon−resonant(s). (5)

The PS and CMS only need the input of a fixed width and are therefore more convenient.

Numerically the difference between the CMS and the PS, even at thresholds, has been shown

to be below 0.1% [20]. The PS requires the separation of the differential cross section into

resonant and non-resonant piece after completing the loop calculation. The CMS requires

the analytic continuation of internal masses appeared in loop integrals into the complex

plane. We choose the CMS here because most of the analytic continuation has already been

provided in the early literature. Moreover, the CMS generally yields smooth distributions

in resonant regions. The only complication is an extra analytic continuation for two-point

scalar integrals, since the renormalization procedure in the CMS requires complex masses

to be assigned to the originally real-valued momenta.

III. THE NLO EW CORRECTION TO THE DY PROCESS

We discuss here the technical details of our calculation of the NLO EW corrections to

lepton-pair production. We begin by separating the gauge-invariant QED subset of the

correction from the full EW result. The photonic corrections can be further split into three

distinct terms: initial-state terms, final-state terms, and initial-final interference terms. Each

piece is identified through coupling combinations of quark and lepton electric charges: Q2q,

Q2l and QqQl respectively. We rederive the photonic correction with finite lepton mass and

5

zero lepton mass separately, and cross-check that the results agree under appropriate choice

of electromagnetic calorimeter setting if the lepton mass is small. The weak portion of the

NLO EW correction consists of gauge boson self-energy insertions, weak vertex corrections,

and ZZ and WW box diagrams. All fermions except for the top quark are taken to be

massless in the weak corrections.

A. QED corrections

The virtual photonic corrections to vertices as well as the fermion wave function renor-

malization constant contain both ultraviolet (UV) and infrared (IR) divergences. QED

gauge invariance guarantees the cancellation of UV singularities between them, so that no

renormalization procedure is required for the one-loop QED correction. The soft singularity

is cancelled completely by the real radiation process qq → llγ.

Due to the assumption of massless quarks in the initial state, a collinear singularity arises

from initial-state photon radiation. This can be absorbed into a bare PDF in complete

analogy to gluon emission in QCD. We do not introduce the photonic contribution to PDF

evolution, because it is suppressed by both O(α) and the smallness of the photon PDF itself.

We use the DIS scheme in constructing the PDF counterterm. The difference compared to

the MS scheme is around 10% for the initial state radiation (ISR) contribution [18]. The

final state radiation (FSR) also contains a collinear singularity if the lepton is assumed to be

massless. A proper procedure to combine photon and leptons when two particles travel too

close together leads to a cancellation of this divergence. This always occurs experimentally

for the electron, since it is hard to separate a electron from its collinear photonic radiation in

the electromagnetic calorimeter. For muons, such a separation is possible and the collinear

singularity is instead regulated by the finite lepton mass in the form of α ln(mµ). A strict

isolation cut on a muon will therefore lead to a considerable photonic correction to the DY

cross section. The results for massless and massive leptons are equivalent to each other up

to O(m2l /M

2Z) if the same recombination procedure is applied, as the ln(ml) dependence

cancels.

The virtual box contribution in QED contains only an IR singulary, which cancels the one

arising from ISR-FSR interference terms in the real radiation process. We use dimensional

regularization to regulate both UV and IR poles. Integration-by-parts (IBP) identities are

used in the loop integration to reduce all tensor integrals to a few master scalar integrals

via fully automated computer algorithms [29]. All relevant scalar integrals with complex

masses are calculated by hand and checked against the existing literature [30–33].

The soft and collinear singularities in the real radiation diagrams are extracted using a

modified two cut-off phase space slicing method [34]. The first cut-off δs defines the soft

region by Eγ < δs√s/2, where s denotes the partonic center of mass energy squared. The

second cut-off δc defines the collinear regions with respect to each fermion (incoming quarks

6

and outgoing leptons) by sfγ < δcs. By approximating the matrix element in the soft

and collinear limits and integrating over only the soft and collinear phase space regions,

we can obtain the IR singularities analytically as 1/ǫ poles in dimensional regularization.

The remaining hard non-collinear phase space can be integrated numerically in four dimen-

sions. Given sufficiently small δs and δc parameters (terms of O(δs), O(δc) and O(δc/δs) are

dropped), the sum of soft, collinear and hard non-collinear contributions will be independent

of the cut-offs. A simply illustration of the cut-off method is as follows:

∫ 1

0

f(x)

x1+ǫ=

∫ δ

0

f(x)

x1+ǫ+

∫ 1

δ

f(x)

x1+ǫ

=

∫ δ

0

f(0)

x1+ǫ+

∫ 1

δ

f(x)

x+O(ǫ) +O(δ)

=

[

−f(0)

ǫ+ f(0) ln(δ)

]

+

∫ 1

δ

f(x)

x,

where we have dropped the O(ǫ) and O(δ) terms as they are infinitesimal parameters. The

first term corresponds to soft and collinear contributions which are regulated via dimensional

regularization. The second term corresponds to the hard non-collinear contribution which

is limited by the cut-off near the phase space boundary. We can easily conclude from

observation of the above formula that extremely small cut-off parameters would lead to

numerical instability. To avoid such issues and reduce artificial parameter dependence in

FEWZ, we further work out the analytic dependence of the cut-off parameters by introducing

counterterms, shown in the example below:

∫ x0

0

f(x)

x1+ǫ=

[

−f(0)

ǫ+ f(0) ln(δ)

]

+

∫ x0

δ

f(x)

x

=

[

−f(0)

ǫ+ f(0) ln(δ)

]

+

∫ x0

δ

f(x)

x−∫ x0

δ

f(0)

x+

∫ x0

δ

f(0)

x

=

[

−f(0)

ǫ+ f(0) ln(δ)

]

+

∫ x0

0

f(x)

x−∫ x0

0

(

f(0)

x− f(0)

x0

ln(x0

δ)

)

.

The third term explicitly depends on the cut-off parameter and is our desired counterterm. It

essentially becomes a subtraction method [35, 36]. The approximate matrix elements in the

soft and collinear regions are integrated over 2-body phase space and cancel IR singularities

from virtual diagrams, which can be interpreted as converting the 1/ǫ pole in dimensional

regulation to the cut-off regulator. The corresponding counterterms render the integrand

of the 3-body phase space finite in the soft and collinear limits, and the hard non-collinear

region can be extended to the full phase space. Agreement is found numerically before and

after applying the subtraction procedure and the dependence on cut-off parameters indeed

vanishes. The derivation of the couterterms can be fully automated after choosing a proper

parameterization of the 3-body phase space, which is done in four space-time dimensions as

the cut-off now regulates IR singularities.

7

B. Weak Correction

The weak correction is decomposed into self-energy insertions of γγ, ZZ and γZ mixing

contributions, corrections to the llγ/Z and qqγ/Z vertices, and ZZ and WW box contribu-

tions. The leading contribution comes from two pieces: the running of the electromagnetic

coupling from the scale Q = 0 to the scale of the hard interaction, and the isospin viola-

tion due to the large mass splitting between top and bottom quarks. The former receives

contributions largely from light fermion loops in the photon self energy insertion Πf 6=t(M2Z).

The latter can be mostly accounted for by the difference between the self-energy insertions

of the W and Z bosons, ∆ρ. We adopt most of the results for the weak correction directly

from Refs. [20] and [37]. The renormalization of the vertices and self energy insertions in

the CMS is described in detail in Section 3.3 of Ref. [20]. Because we implement only the

α(MZ) and Gµ input parameter schemes, the logarithmic dependence on the small fermion

mass cancels out in the coupling renormalization, as demonstrated below.

• α(MZ) scheme:

δgffγ/Zgffγ/Z

⊃ δe(MZ)

e(MZ)= δZe −

1

2∆α(MZ)

= −sWcW

ΣγZT (0)

µ2Z

+1

2Re

{

ΣγγT,f 6=t(M

2Z)

M2Z

}

; (6)

• Gµ scheme:

δgffγ/Zgffγ/Z

⊃ δeGµ

eGµ

= δZe −1

2∆r

= −sWcW

ΣγZT (0)

µ2Z

+c2W2s2W

(

ΣZZT (µ2

Z)

µ2Z

− ΣWT (µ2

W )

µ2W

)

− ΣWT (0)− ΣW

T (µ2W )

2µ2W

−cWsW

ΣγZT (0)

µ2Z

− α

8πs2W

(

6 +7− 4s2W2s2W

ln(c2W )

)

. (7)

Both behave well in the limit of massless fermions. We have suppressed chirality indices in

our notation for simplicity. Another logarithmic dependence on light fermion mass comes

from the photon and Z boson wave function renormalization constants. However, they

appear in both the vertex and self-energy renormalizations and nicely cancel each other.

We can therefore safely neglect all light quark and lepton masses in the computation of the

weak correction. Because the top quark remains the only massive particle, the CKM matrix

can be treated as unity and does not need to be renormalized.

Another complication of renormalization in the CMS is due to the analytic continuation of

the complex momentum appearing in two-point scalar integrals, required in the calculation of

gauge boson self energy insertions. An alternative procedure has been proposed in Ref. [39]

8

based on the expansion around real-valued masses. The error from the expansion is of

O(ΓZ/MZ) for the Z boson self-energy insertion. Special care must to be taken for charged

or colored particles like the W boson. The expansions break down in the presence of photon

or gluon exchange, which generates terms like (s − µ2W ) ln(s − µ2

W ). We instead adopt the

direct approach of performing the analytic continuation. We first note that except for light

fermions whose masses are assumed to be zero, all particles in the W/Z boson self-energy

graphs have masses greater than or equal to MW/Z . Since the internal propagators of these

particles cannot go on-shell when the incoming energy is at the mass of the W or Z boson,

there should be no imaginary part from the loop integration. This implies that no branch

cut is crossed in the integration, and the analytic continuation can be performed using the

analytic expression derived for real-valued momenta. In contrast, massless fermions running

in the self-energy graphs go on-shell for any time-like momentum, and the integration contour

moves away from the branch cut in the direction specified by the Feynman prescription. A

complex mass for the incoming momentum squared leads to the opposite pole treatment

from the Feynman prescription. As we will show later, a simple addition of 2πi to the

original result will fix this issue. A detailed derivation can be found in the Appendix A.

C. Higher-order QCD corrections in the CMS

The application of the CMS, and specifically the complex coupling, leads to a different

LO expression than the original result implemented in FEWZ. Although the difference is

numerically small, we modify the QCD corrections in FEWZ in order to achieve theoretical

consistency. In the limit of massless leptons, the QCD correction to dilepton production can

be schematically written as:

dσγγ

pp→ll∼ Q2

uQ2l |Mu|2 +Q2

dQ2l |Md|2 + 2QuQdQ

2lRe[MuM∗

d]

s2,

dσγZ

pp→ll∼ 2

∑

σ,σ′,τ=+,−

{

QuQl

sRe

[

gσuuZgτllZ

s− µ2Z

Mστu Mσ′τ∗

u

]

+QdQl

sRe

[

gσddZ

gτllZ

s− µ2Z

Mστd Mσ′τ∗

d

]

+QuQl

sRe

[

gσddZ

gτllZ

Mσ′τ∗u Mστ

d

s− µ2Z

]

+QdQl

sRe

[

gσuuZgτllZ

Mστu Mσ′τ∗

d

s− µ2Z

]}

,

dσZZpp→ll ∼

∑

σ,τ=+,−

{ |gσuuZ|2|gτllZ|2|s− µ2

Z|2|Mστ

u |2 + |gσddZ

|2|gτllZ|2

|s− µ2Z|2

|Mστd |2

}

+2∑

σ 6=σ′,τ=+,−

|gτllZ

|2|s− µ2

Z|2{

Re[gσuuZgσ′∗uuZMστ

u Mσ′τ∗u ] +Re[gσddZg

σ′∗ddZMστ

d Mσ′τ∗d ]

}

+2∑

σ,σ′,τ=+,−

|gτllZ

|2|s− µ2

Z |2Re[gσuuZg

σ′∗ddZMστ

u Mσ′τ∗d ], (8)

in which dσγγ

pp→ll, dσZZ

pp→lland dσγZ

pp→llare the contributions from photon, Z and photon-Z

9

interference channels. Mu and Md are matrix elements connected to the photon or Z prop-

agator through up-type quark and d-type quark vertices respectively, σ and σ′ are chirality

indices of the relevant quarks, and τ is the chirality index of the leptons. This structure is

illustrated in Fig. (1) where the incoming parton line, though plotted as fermion line, could

represent either a quark or a gluon. We have abused the chirality index so that a plus(minus)

i = q, q, g

j = q, q, g

u/d

u/d

γ/Z

l−

l+

gσ gτ∝ Mστ ;ij

u/d

FIG. 1: Graphical representation of the QCD corrections to lepton pair production.

index means the interacting fermion and anti-fermion in the vertex are right(left)-handed

and left(right)-handed separately due to helicity conservation. This definition of the matrix

element incorporates all QCD corrections to the initial state. We note that the chirality

indices of the quarks to which the vector boson couples are not necessarily related to the

polarizations of the incoming partons labeled as i and j. All chirality combinations are

summed if the indices are suppressed, and the convolution with appropriate PDFs is omit-

ted for simplicity. The full result with all its structure intact can be restored by the following

substitutions:

|Mu/d|2 =∑

σ,τ=+,−

|Mστu/d|2 + 2

∑

σ 6=σ′,τ=+,−

Re[Mστu/dMσ′τ∗

u/d ],

Re[MuM∗d] =

∑

σ,σ′,τ=+,−

Re[Mστu Mσ′τ∗

d ],

|Mστu/d|2 →

∑

i,j=q,q,g

|Mστ ;iju/d (x1, x2)|2fi(x1)fj(x2),

Re[Mστu/dMσ′τ∗

u/d ] →∑

i,j=q,q,g

Re[Mστ ;iju/d (x1, x2)Mσ′τ ;ij∗

u/d (x1, x2)]fi(x1)fj(x2),

Re[Mστu Mσ′τ∗

d ] →∑

i,j=q,q,g

Re[Mστ ;iju (x1, x2)Mσ′τ ;ij∗

d (x1, x2)]fi(x1)fj(x2). (9)

10

In the original FEWZ, combinations of charge, vector and axial couplings were multiplied

by the corresponding PDF and γ/Z propagator, forming a luminosity function, which was

then convoluted with the squared matrix elements MστqiMσ′τ ′∗

qj. Here σ, σ′, τ and τ ′ denote

vector or axial couplings. Only the real part was kept for the squared matrix element since

the luminosity function is real-valued (except for γZ interference terms due to the width in

the Z propagator; the imaginary part was neglected based on the same reason as explained

below for the CMS). This is no longer true in the CMS. However, we can calculate the

luminosity function with complex numbers, and take the real part in the end. By effectively

neglecting the imaginary part of the squared matrix element, we change the cross section

by O(ΓW/Z/MW/Z) ≈ O(α). The cross section at the lowest order does not change because

its matrix element is always real. We conclude that the QCD part of FEWZ is changed

from the correct formula only by O(αsα), which is beyond our approximation and can be

safely neglected. In practice, we can apply complex couplings in the QCD module of FEWZ

by declaring the coupling constants as complex numbers and modifying the following EW

coupling combinations:

(

gσuuZ/ddZ/llZ

)2

→∣

∣

∣gσuuZ/ddZ/llZ

∣

∣

∣

2

,

gσuuZ/ddZgσ′

uuZ/ddZ → Re[

gσuuZ/ddZgσ′∗uuZ/ddZ

]

. (10)

The electric charges are always real and therefore no operation is necessary for the γγ and

γZ interference channels. The modification is not required if proper conjugation of complex

coupling is carried out from the beginning of the derivation. Alternatively, we can implement

it in terms of vector and axial coupling as follows:

(

gV/A

uuZ/ddZ/llZ

)2

→∣

∣

∣gV/A

uuZ/ddZ/llZ

∣

∣

∣

2

,

gVuuZ/ddZ/llZgAuuZ/ddZ/llZ → Re

[

gVuuZ/ddZ/gA∗uuZ/ddZ/llZ

]

. (11)

IV. PHENOMENOLOGICAL RESULTS

We are now ready to present numerical results using the updated FEWZ simulation

code. We split our phenomenological results into three sections. We first provide a detailed

comparison of the EW corrections implemented in our code with previous calculations in the

literature, to demonstrate consistency between them. We then present several representative

distributions that illustrate the combined NNLO QCD and NLO EW corrections obtained

using FEWZ. In the last part we study distributions of the radiated photon, and point out

several interesting kinematic features that should be observable in LHC data.

11

A. Comparison with the previous literature

We begin our presentation of phenomenological results by cross-checking our EW correc-

tions against the results presented in Ref. [20]. We take the top quark mass to be mt = 174.6

GeV and the Higgs boson mass mH = 115 GeV. The Gµ input parameter scheme is chosen

and the MRST2004QED PDF set is used to include the contribution of the photon PDF.

In the presence of photon radiation, the following recombination procedure is applied be-

fore any acceptance cuts on leptons are implemented. Photons with rapidity |ηγ | > 3 are

discarded as beam remnants. The separation between surviving photons and each lepton,

∆Rl±γ =√

∆η2l±γ +∆φ2l±γ, is calculated, and the photon is recombined with the closest

(anti-)lepton if ∆Rl±γ < 0.1. The following acceptance cuts on leptons are then applied:

pT,l± > 25 GeV, |ηl±| < 2.5.

Mll/GeV > 50 > 100 > 200 > 500 > 1000 > 2000

LO(DH)/pb 738.733(6) 32.7236(3) 1.48479(1) 0.0809420(6) 0.00679953(3) 0.000303744(1)

LO0/pb 738.789(9) 32.723(4) 1.483(1) 0.0809449(8) 0.0067993(6) 0.0003038(1)

LOµ/pb 738.769(9) 32.728(4) 1.483(1) 0.0809451(8) 0.0067993(6) 0.0003037(1)

δγγ,LO(DH)/% 0.17 1.15 4.30 4.92 5.21 6.17

δγγ,LO/% 0.17 1.15 4.30 4.92 5.21 6.18

δQED,rec(DH)/% -1.81 -4.71 -2.92 -3.36 -4.24 -5.66

δQED,rec0 /% -1.79 -4.80 -2.94 -3.41 -4.33 -5.81

δQED,recµ /% -1.77 -4.78 -2.93 -3.41 -4.33 -5.83

δQEDµ (DH)/% -3.34 -8.85 -5.72 -7.05 -9.02 -12.08

δQEDµ /% -3.38 -9.09 -5.85 -7.22 -9.28 -12.47

δweak(DH)/% -0.71 -1.02 -0.14 -2.38 -5.87 -11.12

δweak/% -0.70 -1.02 -0.14 -2.38 -5.87 -11.11

TABLE I: Cross sections of dilepton production at a 14 TeV LHC using MRST2004QED PDFs,

for various cuts on the dilepton invariant mass. The content of each row is explained in detail in

the text.

Tables I and II summarize the comparisons between our results and those of Ref. [20],

denoted by DH. The subscript 0 denotes results obtained in the massless lepton mode, while

the subscript µ indicates results obtained using the muon mass. When the recombination

procedure is applied, the results given by the muon mode and massless lepton mode should

be identical, since the large logarithms α ln(m2l /M

2Z) in the FSR and virtual photonic contri-

butions cancel . Separate results for muons without recombination are given for comparison.

The weak correction is generally small. For the LHC, it is enhanced in the high mass tail due

to large EW Sudakov logarithms, reaching the same order as QCD and QED corrections.

Results of photon-induced dilepton production are also listed, and are very small due to the

suppression of photon PDF and the lack of resonance structure in this channel. However,

12

Mll/GeV > 50 > 100 > 150 > 200 > 400 > 600

LO(DH)/pb 142.7878(7) 6.62280(3) 0.824114(3) 0.294199(1) 0.01775063(5) 0.001778465(5)

LO0/pb 142.783(1) 6.6215(8) 0.8241(3) 0.294195(2) 0.017750(2) 0.0017789(6)

LOµ/pb 142.783(1) 6.6225(8) 0.8246(3) 0.294197(2) 0.017749(2) 0.017785 (6)

δγγ,LO(DH)/% 0.15 0.72 1.54 1.44 0.83 0.57

δγγ,LO/% 0.15 0.72 1.54 1.44 0.83 0.57

δQED,rec(DH)/% -1.85 -4.87 -3.65 -3.83 -5.16 -6.56

δQED,rec0 /% -1.82 -4.96 -3.70 -3.89 -5.29 -6.76

δQED,recµ /% -1.80 -4.93 -3.68 -3.88 -5.28 -6.75

δQEDµ (DH)/% -3.44 -8.93 -6.46 -6.86 -9.56 -12.42

δQEDµ /% -3.47 -9.15 -6.59 -7.02 -9.84 -12.83

δweak(DH)/% -0.70 -1.01 -0.12 -0.15 -1.25 -2.60

δweak/% -0.70 -1.00 -0.13 -0.15 -1.25 -2.60

TABLE II: Cross sections of dilepton production at the 1.96 TeV Tevatron using MRST2004QED

PDFs, for various cuts on the dilepton invariant mass. The content of each row is explained in

detail in the text.

its contribution can reach 5% in the high energy range as pointed out in Ref. [20]. We find

agreement with the numerical results of Ref. [20]. The small difference at the 0.1% level for

the photonic correction can be explained by the inclusion of the photon PDF in a redefined

quark PDF, which was performed in Ref. [20] but not here.

Fig. 2 uses the histogramming feature of FEWZ to reproduce the percent changes in

various distributions arising from QED and weak corrections in the Z resonance region.

We show several different lepton identification scenarios. An ideal resolution is assumed in

the bare muon case, and no recombination procedure is applied. For massless leptons, two

values of separation ∆Rlγ are considered for the recombination procedure. A larger ∆Rlγ

effectively lowers the detector resolution and leads to a more inclusive observable. The

photonic correction is dominated by FSR, which is particularly sensitive to the choice of the

recombination procedure.

The invariant mass of the lepton pair Mll receives a positive photonic correction below the

Z peak and a negative correction above the Z peak, leading to a shift of the Z resonance

position to a smaller value. The weak correction has a similar feature, but its size is an

order of magnitude smaller in the resonance region. The lepton pT distribution has a well-

known Jacobian peak at MZ/2, which is also distorted primarily by the photonic correction.

We note that a proper description of the pT,l distribution near the Jacobian peak requires

resummation of multiple soft photon effects. The bottom two histograms in Fig. 2 are the

rapidity distributions of the reconstructed Z and of the lepton. The NLO EW correction to

the Z rapidity is roughly constant except at the boundary of phase space. The contribution of

photon-induced dilepton production is highly suppressed by the photon PDF and is negligible

in most situations. It grows relatively more important in the high rapidity region due

13

FIG. 2: Percentage change of various components of the NLO EW correction, with respect to the

leading-order result. The MRST2004QED set is used, and the LHC center-of-mass energy is taken

to be 14 TeV. Clockwise from the upper left, the plots show the lepton-pair invariant mass, the

lepton-pair rapidity, the lepton pT , and the lepton pseudorapidity

to the nature of its t-channel exchange diagrams. All histograms are consistent with the

distribution shapes given in Ref. [20]. The extensive comparison presented above reveals

good agreement between our results and previous work.

B. Combination of electroweak and QCD corrections in FEWZ

Having validated our calculation, we next study the combination of the NLO EW and the

NNLO QCD corrections. We switch to√s = 8 TeV in this subsection. In Fig. (3), we display

the full corrections to dilepton production for several observables as a demonstration of the

full functionality of FEWZ. The NLO EW correction, as explained above, plays an important

role in defining the distribution shapes near the Z resonance. The shift of events from the

Z-peak to lower invariant masses caused by FSR effects is apparent in the uppermost left

plot. The large EW Sudakov logarithms cause the rise in the high-mass tail of the invariant-

14

FIG. 3: Several representative distributions showing the combination of NNLO QCD and NLO

EW corrections to lepton-pair production at a√s = 8 TeV LHC. The MSTW2008NNLO PDF

set has been used, with the hatched regions corresponding to the estimated 68% PDF error. The

αS error has not been included. Clockwise from the upper left, the plots show the lepton-pair

invariant mass, the lepton-pair rapidity, the lepton pT , and the lepton pseudorapidity.

mass distribution seen in the plot. We note that because of the cut pT,l > 25 GeV, the

invariant mass of the lepton pair is restricted to Mll > 50 GeV, coinciding exactly with

the cut on this variable that we impose. Sensitivity to this phase-space restriction leads to

the large shift from the leading-prediction near this boundary. The lepton pT distribution

in the lower-left panel exhibits the usual Jacobian peak at MZ/2. Resummation of soft-

photon and soft-gluon effects is needed for a proper description near this boundary. Both

the lepton pseudorapidity and lepton-pair rapidity in the rich panels show little sensitivity

to higher-order effects.

15

C. Distributions of photon radiation in Drell-Yan production

We proceed to examine distributions of photon radiation at the LHC. To define photon

experimentally, we impose the following cuts in our analysis:

pT,γ > 20GeV, |ηγ| < 2.5, ∆Rlγ > 0.05. (12)

We note that the experimental capabilities permit a much lower cut on photon pT , reaching

down to 5 GeV, but we set the cut high here to reveal certain kinematic features in the

prediction. We focus on the invariant-mass region below the Z-peak, where FSR effects are

enhanced. The following cuts are also applied in our study:

phardT,l > 30GeV, 30GeV < Mll < 86GeV,

psoftT,l > 10GeV, |ηl| < 2.5. (13)

These cuts are motivated by an ongoing study of photon radiation in the Drell-Yan process

within CMS [38]. We also study distributions where the softer-lepton pT cut is increased to

psoftT,l > 20GeV. To facilitate comparison with the ongoing experimental study we consider

the√s = 7 TeV LHC.

FIG. 4: Distributions of photon radiation in the Drell-Yan process. The left panel shows the

distribution of separation between the photon and the nearest leptons using both ∆φ and ∆R as

a distance measure. The right panel shows the photon transverse momentum spectrum. The cuts

are those described in Eqs. (12) and (13).

The left histogram in Fig. 4 shows the distribution of ∆Rlγ =√

(∆ηlγ)2 + (∆φlγ)2,

defined as the separation between the photon and the nearest lepton. It peaks toward zero

separation, corresponding to the FSR collinear divergence. We also present the distribution

of ∆φlγ, the difference in azimuthal angle between the photon and the nearest lepton, in the

same histogram. There is a kink in both the ∆Rlγ and ∆φlγ distributions, the location of

which is roughly given by π−arccos(pT,γ,min/MZ) ≈ π/2 for small pT,γ,min. It occurs for the

maximum ∆φlγ when the two leptons and the photon travel with the same pseudorapidity.

16

The exact location of the feature is at the angle π − arccos(pT,γ,min/√

M2Z + p2T,γ,min) for

ISR photons and π − arccos(pT,γ,min/(MZ − pT,γ,min)) for FSR photons. At small pT,γ,min,

the two expressions yields roughly the same number. However, the dominance of FSR

photons with small separation washes out this effect. At large pT,γ,min, the collinear FSR

contribution becomes suppressed and the prominence of the effect increases. We can enhance

the kinematic feature by increasing the cut on the softer lepton, which further reduces the

collinear FSR contribution. This is shown in Fig. 5, where we have used psoftT,l > 20GeV.

The kink has now become a small peak.

FIG. 5: The separation between the photon and the nearest lepton using instead psoftT,l > 20GeV.

The photon pT distribution in the right panel of Fig. 4 has a shoulder at around 35 GeV.

It can be explained by noting that the majority of photon radiation occurs as a result of

bremsstrahlung radiation off a lepton from an on-shell Z. Most of the photon radiation is

collinear to the direction of the lepton, and is limited to a maximum pT of half of the Z mass

as it recoils against the other lepton. However, the requirement of a lepton with a pT greater

than 10 GeV reduces the maximally allowed pT for FSR photon to MZ/2−pT,l,min ≈ 35GeV

where pT,l,min = 10 GeV. We can interpret the small bump as the “Jacobian peak” of the

photon, in analogy to the lepton case.

V. CONCLUSION

In this paper we have presented a combination of the NNLO QCD and NLO EW cor-

rections to the Drell-Yan production of a lepton pair. This combination was not previously

available in the literature. We have incorporated these corrections into an updated version of

the analysis code FEWZ, which is used in numerous experimental studies. The combination

17

of both the QCD and EW sources of higher-order corrections in a single program elimi-

nates the need to piece them consistently. We have recalculated the EW corrections in the

complex-mass scheme and have cross-checked our results in detail against previous results

in the literature. We have presented several new techniques in the course of our calculation,

including a combination of phase-space slicing and subtraction methods for handling real-

emission corrections, and a treatment for the analytic continuation of two-point functions

needed in the complex-mass scheme.

We have presented numerous phenomenological results relevant for LHC studies, including

the first theoretical results for distributions containing both NNLO QCD and NLO EW

effects. We have also discussed several interesting kinematic features in photon-radiation

distributions caused by imposing experimental cuts. These should be observable in the LHC

data, and would offer interesting tests of the underlying mechanism for producing photons

in association with a lepton pair in hadronic collisions.

Acknowledgements

We are grateful to A. Kubik, M. Schmitt and S. Stoynev for useful discussions on the on-

going experimental studies at CMS. We also thank S. Quackenbush for advice and help in

finalizing the new version of FEWZ. This work is supported by the U.S. Department of En-

ergy, Division of High Energy Physics, under contract DE-AC02-06CH11357 and the grants

DE-FG02-95ER40896 and DE-FG02-08ER4153, and with funds provided by Northwestern

University.

Appendix A: Analytic Continuation of the Scalar Bubble Integral

The two-point one-loop scalar integral (the bubble integral) is defined as follows:

B0(s;M21 ,M

22 ) =

(2πµ)4−D

iπ2

∫

dDl1

(l2 −M21 + iε)((l + p)2 −M2

2 + iε), (A1)

where s = p2. The space-time dimension is taken to be D = 4− 2ǫ. Introducing a standard

Feynman parameterization of the integral, we obtain

B0(s;M21 ,M

22 ) = (4πµ2)ǫΓ(ǫ)

∫ 1

0

dx [−x(1 − x)s+ xM22 + (1− x)M2

1 − iε]−ǫ

= (4πµ2)ǫ{

1

ǫ−∫ 1

0

dx ln(−x(1 − x)s+ xM22 + (1− x)M2

1 − iε)

}

+O(ǫ).

(A2)

The logarithm is defined to have a branch cut on the negative real axis. If s, M21 , and

M22 are real parameters, the integration contour stays below the branch cut due to the

18

Feynman prescription for the propagators. If we replace M21 and M2

2 with the complex

masses µ21,2 = M2

1,2− iΓ1,2M1,2, the contour simply moves further away from the branch cut.

This can change if s is also a complex mass, as shown in Fig. 6. If the masses M1 and M2

are sufficiently large so that both particles in the loop cannot go on shell, then the argument

inside the logarithm of the integrand always has a positive real part, and the contour never

crosses the branch cut. If the particles in the loop can go on-shell, the branch cut can be

crossed.

M21

M22

−Γ1M1

−Γ2M2

F (s, µ21, µ

22)

FIG. 6: The contour for lnF in the complex plane, assuming arbitrary complex parameters.

For the computation of the gauge boson self-energy insertions, the particles running in

the loop are either massless fermions, a heavy top quark, a Higgs boson or the gauge boson

themselves. Except for massless fermions, the other internal particles can never be on-shell.

Therefore, we only need to worry about the bubble integral of the form B0(s, 0, 0) with

s = µ2Z or µ2

W . It is easy to work out the integral assuming s is the complex mass of either

the W or Z boson:

B0(s; 0, 0) = (4πµ2)ǫ{

1

ǫ−

∫ 1

0

dx ln(−x(1 − x)s− iε)

}

+O(ǫ)

= (4πµ2)ǫ{

1

ǫ+ 2− [ln(s− iε) + ln(−1 − iε/s)]

}

+O(ǫ). (A3)

In the last line, we have written the integral in a form valid for both real s, or if s is given

by a complex mass. The second logarithm consistently yields −iπ as long as Re(s) > 0,

ensuring a smooth transition for a complex mass valued s. It is easy to find the following

Taylor expansion assuming a small imaginary part for s:

B0(µ2W/Z ; 0, 0) = B0(M

2W/Z ; 0, 0) + (µ2

W/Z −M2W/Z)

dB0(s; 0, 0)

ds

∣

∣

∣

∣

s=M2

W/Z

+O(Γ2W/Z/M

2W/Z),

(A4)

19

which demonstrates the equivalence at NLO between our result and the alternative procedure

proposed in Ref. [39].

[1] S. D. Drell and T. -M. Yan, Phys. Rev. Lett. 25, 316 (1970) [Erratum-ibid. 25, 902 (1970)].

[2] S. Haywood et al., arXiv:hep-ph/0003275.

[3] M. Dittmar, F. Pauss and D. Zurcher, Phys. Rev. D 56, 7284 (1997) [arXiv:hep-ex/9705004];

V. A. Khoze, A. D. Martin, R. Orava and M. G. Ryskin, Eur. Phys. J. C 19, 313 (2001)

[arXiv:hep-ph/0010163]; W. T. Giele and S. A. Keller, arXiv:hep-ph/0104053.

[4] S. Chatrchyan et al. [CMS Collaboration], Phys. Rev. D 84, 112002 (2011) [arXiv:1110.2682

[hep-ex]].

[5] For recent studies, see R. D. Ball, L. Del Debbio, S. Forte, A. Guffanti, J. I. Latorre, J. Rojo

and M. Ubiali, Nucl. Phys. B 838, 136 (2010) [arXiv:1002.4407 [hep-ph]]; R. S. Thorne,

A. D. Martin, W. J. Stirling and G. Watt, arXiv:1006.2753 [hep-ph]; H. L. Lai, M. Guzzi,

J. Huston, Z. Li, P. M. Nadolsky, J. Pumplin and C. P. Yuan, arXiv:1007.2241 [hep-ph];

S. Alekhin, J. Blumlein and S. Moch, arXiv:1202.2281 [hep-ph].

[6] R. Hamberg, W. L. van Neerven and T. Matsuura, Nucl. Phys. B 359, 343 (1991) [Erratum-

ibid. B 644, 403 (2002)].

[7] C. Anastasiou, L. J. Dixon, K. Melnikov and F. Petriello, Phys. Rev. Lett. 91, 182002 (2003)

[hep-ph/0306192].

[8] C. Anastasiou, L. J. Dixon, K. Melnikov and F. Petriello, Phys. Rev. D69, 094008 (2004)

[hep-ph/0312266].

[9] K. Melnikov and F. Petriello, Phys. Rev. Lett. 96, 231803 (2006) [arXiv:hep-ph/0603182].

[10] K. Melnikov and F. Petriello, Phys. Rev. D 74, 114017 (2006) [arXiv:hep-ph/0609070].

[11] S. Catani, L. Cieri, G. Ferrera, D. de Florian and M. Grazzini, Phys. Rev. Lett. 103, 082001

(2009) [arXiv:0903.2120 [hep-ph]].

[12] S. Catani, G. Ferrera and M. Grazzini, JHEP 1005, 006 (2010) [arXiv:1002.3115 [hep-ph]].

[13] R. Gavin, Y. Li, F. Petriello and S. Quackenbush, Comput. Phys. Commun. 182, 2388 (2011)

[arXiv:1011.3540 [hep-ph]].

[14] R. Gavin, Y. Li, F. Petriello and S. Quackenbush, arXiv:1201.5896 [hep-ph].

[15] C. M. Carloni Calame, G. Montagna, O. Nicrosini and A. Vicini, JHEP 0710, 109 (2007)

[arXiv:0710.1722 [hep-ph]].

[16] F. A. Berends and R. Kleiss, Z. Phys. C 27, 365 (1985).

[17] F. A. Berends, R. Kleiss, J. P. Revol and J. P. Vialle, Z. Phys. C 27, 155 (1985).

[18] U. Baur, S. Keller and W. K. Sakumoto, Phys. Rev. D 57, 199 (1998) [hep-ph/9707301].

[19] U. Baur, O. Brein, W. Hollik, C. Schappacher and D. Wackeroth, Phys. Rev. D 65, 033007

(2002) [hep-ph/0108274].

[20] S. Dittmaier and M. Huber, JHEP 1001, 060 (2010) [arXiv:0911.2329 [hep-ph]].

[21] M. Ciafaloni, P. Ciafaloni and D. Comelli, Phys. Rev. Lett. 84, 4810 (2000)

[22] P. Ciafaloni and D. Comelli, Phys. Lett. B 446, 278 (1999); Phys. Lett. B 476, 49 (2000)

[23] V. S. Fadin, L. N. Lipatov, A. D. Martin and M. Melles, Phys. Rev. D 61, 094002 (2000)

[24] J. H. Kuhn, A. A. Penin and V. A. Smirnov, Eur. Phys. J. C 17, 97 (2000)

[25] C. Bernaciak and D. Wackeroth, arXiv:1201.4804 [hep-ph].

[26] L. Barze, G. Montagna, P. Nason, O. Nicrosini and F. Piccinini, JHEP 1204, 037 (2012)

20

[arXiv:1202.0465 [hep-ph]].

[27] A. Denner and S. Dittmaier, Nucl. Phys. Proc. Suppl. 160, 22 (2006) [hep-ph/0605312].

[28] S. Dittmaier and A. Kaiser, Phys. Rev. D 65, 113003 (2002) [hep-ph/0203120].

[29] C. Anastasiou and A. Lazopoulos, JHEP 0407, 046 (2004) [arXiv:hep-ph/0404258].

[30] G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B 153, 365 (1979).

[31] T. Hahn and M. Perez-Victoria, Comput. Phys. Commun. 118, 153 (1999) [hep-ph/9807565].

[32] R. K. Ellis and G. Zanderighi, JHEP 0802, 002 (2008) [arXiv:0712.1851 [hep-ph]].

[33] A. Denner and S. Dittmaier, Nucl. Phys. B 844, 199 (2011) [arXiv:1005.2076 [hep-ph]].

[34] B. W. Harris and J. F. Owens, Phys. Rev. D 65, 094032 (2002) [arXiv:hep-ph/0102128].

[35] S. Frixione, Z. Kunszt and A. Signer, Nucl. Phys. B 467, 399 (1996) [hep-ph/9512328].

[36] S. Catani and M. H. Seymour, Nucl. Phys. B 485, 291 (1997) [Erratum-ibid. B 510, 503

(1998)] [arXiv:hep-ph/9605323].

[37] A. Denner, Fortsch. Phys. 41, 307 (1993) [arXiv:0709.1075 [hep-ph]].

[38] A. Kubik and M. Schmitt, private communication.

[39] A. Denner, S. Dittmaier, M. Roth and L. H. Wieders, Nucl. Phys. B 724, 247 (2005) [Erratum-

ibid. B 854, 504 (2012)] [hep-ph/0505042].

21